A Fuzzy Algorithm for Solving a Class of Bi-Level Linear Programming Problem

Transcription

1 Appl. Math. Inf. Sc. 8, No. 4, (2014) 1823 Appled Mathematcs & Informaton Scences An Internatonal Journal A Fuzzy Algorthm for Solvng a Class of B-Level Lnear Programmng Problem Lu Zhang School of Economcs and Management, Chna Unversty of Geoscences, Wuhan , Chna Receved: 8 Aug. 2013, Revsed: 10 Nov. 2013, Accepted: 11 Nov Publshed onlne: 1 Jul Abstract: Ths paper proposes a knd of b-level lnear programmng problem, n whch there are two decson makers n a herarchy and they have a common varable. To deal wth ths b-level problem, we ntroduce a vrtual decson maker, who controls the common varable to maxmze the sum of the objectve functons of the upper and lower level decson maker (the leader and follower). To llustrate the partal cooperaton, the vrtual decson maker chooses hs/her decson before the leader because the leader and the follower exchange the nformaton to maxmze ther total benefts. Then the leader chooses hs/her decson before the follower. Consequently, a tr-level programmng model s obtaned. Then, a fuzzy approach s presented to solve ths tr-level programmng. Fnally, a numercal example s solved to demonstrate the feasblty of the model after presentng a fuzzy programmng approach. Keywords: B-level lnear programmng; Common varable; Tr-level programmng; Fuzzy programmng algorthm 1 Introducton The b-level programmng s a nested optmzaton problem wth two levels (namely the upper and lower level) n a herarchy. It s a practcal and useful tool for solvng decson makng (DM) problems wth herarchal structure, and has been used to solve many practcal problems, such as engneerng desgn, management, economc polcy, traffc problem and so on. Therefore, the b-level programmng has been developed and researched by many authors. For the recent surveys and monographs readers can refer to [1,2,3,4,5]. In a b-level programmng, the upper level decson maker (the leader) optmzes hs/her objectve functon ndependently and s affected by the reacton of the lower level decson maker (the follower) who makes hs/her decson after the former. Ther objectve functons generally conflct each other. However, most problems encountered n practce fall nto the stuaton n whch they depend partly on the degree of nteracton or cooperaton between them, although the nformaton between them s ncomplete and vague. So, the decson makers partally cooperate. For nstance, the b-level programmng problem wth a common varable s presented by consderng optmal bddng strateges between the power Sellers (Power Companes) and Buyer (Dstrbuton Center) for contract arrangements of the mddle-term contracts and the spot market transactons under uncertan electrcty spot market [6]. Then, several researchers have also nvestgated ths knd of programmng problem. Lu et al. [7] proposed a comprehensve framework for b-level multfollower programmng (BLMFP) problems and Sh et al proposed an extended KuhnCTucker approach for lnear BLMFP problems wth shared varables among followers[8]. Later, the extended Kth-best approach[9] are also proposed for BLMFP wth shared varables among followers. In the above papers, they ntroduce a thrd party called a vrtual follower: the (K + 1)th follower controls the varable z, so the lnear BLMFP wth partal shared varables among followers s equvalently transformed nto the lnear BLMFP wthout partal shared varables among followers, whch s easly solved. In ths transformaton, the (K + 1)th follower s objectve functon s the sum of the th follower s objectve functon(=1,2,,k). Based on the above researches, for the b-level lnear programmng problem wth a common varable between the leader and the follower, we ntroduce a thrd party called a vrtual decson maker, who controls the the common varable, and hs/her objectve functon s the sum of those of the leader and follower. The dfference Correspondng author e-mal: zhanglcug@163.com

2 1824 L. Zhang: A Fuzzy Algorthm for Solvng a Class of B-Level Lnear... wth the above s that the vrtual decson maker chooses hs/her decson before the leader. Thus, a tr-level programmng model s obtaned, n whch frstly the the vrtual decson maker choose hs/her decson, then the leader chooses hs/her decson before the follower. Our reformaton shows not only the vrtual decson maker denotng the overall objectves s more mportant than the leader and the follower s objectves but also the leader and follower s am to exchange the nformaton s to maxmze the sum of ther objectves. Then, we have to take the algorthm for the above model nto account. The varous tradtonal algorthms to solve the b-level problem can be roughly classfed nto the followng knds[2]: extreme-pont approaches for the lnear case, branch-and-bound, complementary pvotng, descent methods, penalty functon methods, trust-regon methods and so on. However, the b-level programmng s not a convex problem, and not dfferentable anywhere, and t s hard to solve. Jeroslow[10] frstly ponted out that the b-level programmng problem s a NP-Hard problem. Then, Ben-Ayed and Blar[11] and Bard[12] proved sequentally that the b-level lnear programmng problem s a NP-Hard problem and searchng for locally optmal soluton to the b-level lnear programmng s also a NP-Hard problem [13]. See Ref. [14] for more detals about complexty ssues about b-level lnear programmng. Recently, Shh et al [15] developed a fuzzy approach, namely nteractve fuzzy decson makng method, for solvng the b-level programmng problems by usng the concept of tolerance membershp functons and multple objectve decson makng. For adjustng the decson makng process between the dfferent levels and also between the decson makers of the same level, Shh and Lee [16] ntroduced compensatory operators. By usng these compensatory operators, the soluton procedures for the varous types of multple level decson problems are formulated. More nteractve fuzzy decson makng methods have extensvely been appled to b-level and multlevel programmng problems [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Whle, some researchers have proposed fuzzy approaches for solvng the b-level programmng problem wth a common varable among the mult-followers [28, 29, 30]. In ths paper, we descrbe a fuzzy programmng approach on the bass of the above fuzzy decson makng methods. The remanng of our paper s organzed as follows: Secton 2 ntroduces the b-level lnear programmng wth a common varable between the leader and follower. Then, the orgnal b-level model s transformed nto a tr-level programmng model by ntroducng the vrtual decson maker. Secton 3 proposes a fuzzy programmng algorthm to solve the transformed tr-level programmng model. An llustratve example s provded n Secton 4. Secton 5 concludes ths paper. 2 Problem Formulaton In ths paper, we research the followng b-level lnear programmng problem (BLP) mn x,y 0 F 1(x,y,z) (1) where x,z solve the followng problem mn x,z 0 F 2(x,y,z) s.t. G(x,y,z) 0 where F 1 (x,y,z) and F 2 (x,y,z) are the upper and lower level objectve functons, respectvely. G(x, y, z) s the constrant functon. y R n y and z R n z are the decson varables controlled by the upper and lower level decson makers, respectvely. x R nx s the common varable between the leader and the follower. For smplcty, we denote t =(x,y,z). To transform the above BLP, we ntroduce a vrtual decson maker, who control the common varable between the leader and follower. Hs/her objectve functon s the sum of the leader and follower s objectve functons. The vrtual decson maker wll choose hs/her decson before the orgnal leader to llustrate hs/her mportant status because the am that the leader and follower change the nformaton through the common varable s to optmze ther total benefts. Therefore, we can obtan the followng transformed tr-level programmng model (TLP) (T LP) mn x 0 F 0(x,y,z)=F 1 (x,y,z)+f 2 (x,y,z) (The frst level) (2) where y and z solve the followng problem mn F 1(x,y,z) y 0 where z solve the followng problem mn F 2(x,y,z) z 0 s.t. G(x,y,z) 0 (The second level) (The thrd level) Next, some notatons and defntons about tr-level programmng problem are ntroduced: (I)The permssble set of TLP: S={(x,y,z) G(x,y,z) 0} (II)The projecton of S onto the frst level decson space: S 0 (X)={x 0 (y,z),such that (x,y,z) S} (III)The permssble set of the second level programmng for fxed x S 0 (X): S 1 (x)={y 0 z,such that G(x,y,z) 0} (IV)The ratonal reacton set of the second level programmng for fxed x S 0 (X): M 1 (x)={y 0 y argmnf 1 (x,y,z),y S 1 (x)}

3 Appl. Math. Inf. Sc. 8, No. 4, (2014) / (V)The permssble set of the thrd level programmng for fxed x and y: S 2 (x,y)={z 0 G(x,y,z) 0} (VI)The ratonal reacton set of the thrd level programmng for fxed x and y: M 2 (x,y)={z 0 z argmnf 2 (x,y,z),z S 2 (x,y)} (VII)The nducble regon of TLP: IR=(x,y,z) (x,y,z) S,y M 1 (x),z M 2 (x,y) For any tr-level programmng problem, the care must be taken when the soluton to the thrd level programmng s not unque for fxed x and y. The problem of multple optmal soluton to the thrd level programmng can be solved by the smlar method proposed to overcome the problem to b-level programmng problem[31]. Here, to ensure that the problem (2) s well posed, S s assumed to be nonempty and compact, and S 1 (x),s 2 (x,y), M 1 (x), M 2 (x,y) are all nonempty. At the same tme, we consder the stuaton that there s a unque soluton to the thrd level programmng for fxed x and y. Then, the defntons of the feasblty and optmalty for BLP are gven as follows: Defnton 1 (x,y,z) S s called the feasble soluton to the problem TLP f(x,y,z) IR. Defnton 2 (x,y,z ) S s called the optmal soluton to the problem TLP f F 0 (x,y,z ) F 0 (x,y,z), (x,y,z) IR. 3 The fuzzy programmng algorthm To begn wth a fuzzy decson makng process, we obtan the optmal soluton of each decson maker calculated n solaton. If the ndvdual optmal soluton t =(x,y,z ) are the same, then a satsfactory soluton of the system has been reached. However, they are always dfferent because the decson makers wth the conflctng objectve functons behave non-cooperatvely. Therefore, the fuzzy decson makng process begns at the frst level. To obtan a satsfactory soluton, the frst level decson maker should provde hs/her preferred ranges for F 0 and x to the second level decson maker, who has a wder feasble doman to search for hs/her optmal soluton. Frstly, the membershp functons are ntroduced by usng fuzzy set theory[35]. The membershp functons can be lnear, pecewse lnear, exponental, logarthmc, hyperbolc, nverse hyperbolc, quadratc, etc. For smplcty, we use the followng lnear membershp functon for the objectve functons F (=0,1,2)[15]: µ F (F (t))= 1, F (t) F L F U F (t) F U F L, F L F (t) F U 0, F (t) F U (3) where F U,F L are the upper and lower bound of F on S. And the membershp functon for x can be formulated as follows[20]: [x (x1 el x)], x e l 1 el x x x1 x µ x (x)= [(x1 +er x) x] e r, x x 1 x x 1 + er (4) x 0, otherwse where the nterval[x 1 el x,x 1 +er x] denotes the range of the decson of x around x 1, and el x and e r x are the negatve and postve tolerances for x at x 1, respectvely. The second level decson maker searches for the satsfactory soluton to mnmze hs/her objectve functon on the bass of guaranteeng that the frst level decson maker satsfes ths satsfactory soluton. Thus, the frst level decson maker should gve the mnmum acceptable degree of satsfacton β and α for F 0 and x. Hence µ F0 (F 0 ) β and µ x (x) α. Let δ be the mnmum acceptable degree of satsfacton of the second level decson maker. So, µ F1 (F 1 ) δ. To resolve the conflct between these two decson makers and to avod rejecton of satsfactory soluton by the frst level decson maker, the second level decson maker must maxmze α,β and δ. Let λ = mn{α,β,δ}. Thus, the second level auxlary problem s max λ (5) s.t. G(x,y,z) 0 µ F0 (F 0 ) β µ x (x) α µ F1 (F 1 ) δ If the two decson makers are satsfed wth ths soluton, then the thrd level s also ncluded. If not, then they may modfy the tolerance values or may even change the membershp functons and the second level decson maker solves a new auxlary problem agan. The process contnues untl the satsfactory soluton s attaned for top two levels after whch the thrd level s ncluded. Agan both the hgher level decson makers pass ther preferred values of ther decson varables and objectve functons separately to the thrd-level decson maker. As the same as the above procedure, the thrd level decson maker solves the auxlary problem as follows: max λ (6) s.t. G(x,y,z) 0 µ F0 (F 0 ) β µ x (x) α µ F1 (F 1 ) δ µ y (y) γ µ F2 (F 2 ) ε where γ s the second level decson maker s mnmum acceptable degree of satsfacton for y. ε s the thrd level

4 1826 L. Zhang: A Fuzzy Algorthm for Solvng a Class of B-Level Lnear... decson maker s mnmum acceptable degree of satsfacton for F 2. λ = mn{α,β,γ,δ,ε}. And the membershp functon for y can be formulated as follows: µ y (y)= [y (y 2 el y)] e l y, y 2 el y x y 2 [(y 2 +er y) y] e r, y y 2 y y 2 + er y 0, otherwse where the nterval[y 2 el y,y 2 +er y] denotes the range of the decson of y around y 2, and el y and e r y are the negatve and postve tolerances for y at y 2, respectvely. If these three decson makers are satsfed wth ths soluton, the overall satsfactory soluton has been reached. If not, then they may modfy the tolerance values or may even change the membershp functons and the thrd level decson maker solves a new auxlary problem agan. The process contnues untl the satsfactory soluton s attaned for these three decson makers. Furthermore, the p > 3 level programmng problem can be solved by ths fuzzy programmng approach only by ncludng the succeedng lower level nto the system and the process repeats. The process contnues untl the last level s ncluded nto the system. 4 Numercal example In ths secton, we propose a numercal example to llustrate feasblty of the proposed model. (7) mn x,y F 1(x,y,z)= 42x x y 1 10y 2 23z 1 20z 2 (8) mn F 2 (x,y,z)= 39x x y 1 + 9y 2 30z 1 40z 2 x,z s.t. x 1 + 4x 2 27y 1 14y 2 + z 1 20z x 1 35x 2 23y 1 + 2y z z x x 2 9y 1 18y z 1 11z x 1 20x 2 + 6y y z 1 11z x x 2 31y 1 8y 2 15z 1 25z x 1 6x y 1 23y z 1 35z x x 2 25y y z 1 2z x x y 1 13y 2 20z 1 + 7z x 1 29x y y 2 29z 1 38z 2 48 x 0, y 0, z 0, =1,2. where x=(x 1,x 2 ) s the common varable. We ntroduce a vrtual decson maker, who controls the common varable. Then the above model can be transformed nto the followng tr-level programmng problem: mn x F 0 (x,y,z)= 81x x y 1 y 2 53z 1 60z 2 (9) mn y F 1 (x,y,z)= 42x x y 1 10y 2 23z 1 20z 2 mn z F 2 (x,y,z)= 39x x y 1 + 9y 2 30z 1 40z 2 s.t. x 1 + 4x 2 27y 1 14y 2 + z 1 20z x 1 35x 2 23y 1 + 2y z z x x 2 9y 1 18y z 1 11z x 1 20x 2 + 6y y z 1 11z x x 2 31y 1 8y 2 15z 1 25z x 1 6x y 1 23y z 1 35z x x 2 25y y z 1 2z x x y 1 13y 2 20z 1 + 7z x 1 29x y y 2 29z 1 38z 2 48 x 0, y 0, z 0, =1,2. We obtan optmal soluton of each decson maker calculated n solaton by usng Lngo [36]. The results are lsted as follows: F0 = at t1 = (0.6722,1.0220,0.8325,0.7185,0.4484,1.0925). F1 = at t2 = (0.7486,1.2323,0.1969,0.3632,0.0000,1.0175). = at F2 t3 = (0.0455,0.9576,1.3410,0.5328, ,1.5945). Obvously, they can not reach a satsfactory soluton because the solutons of the three decson makers are not the same. We consder the top two level decson makers. To present the membershp functon, the upper and lower bounds for F ( = 0,1) can be calculated by use of Lngo as follows: F0 U = , F0 L = , F1 U = ,FL 1 = The frst level decdes x 1 = wth 0.5(negatve) and 1(postve) tolerance and x 2 = wth 0.5(negatve) and 0.3(postve) tolerance, where these tolerances are subjectvely chosen. The satsfactory soluton (0.6786, , , , , ) can be obtaned by solvng the second level auxlary problem usng Lngo. The two decson makers are both satsfed wth the satsfactory soluton, then the thrd level can be ncluded. The bounds for all objectve functons are calculated by usng Lngo as follows: F0 U = ,F0 L = F1 U = ,F1 L = , F2 U = ,FL 2 = The frst level decdes x 1 = wth 0.6(negatve) and 0.4(postve) tolerance and x 2 = wth 1(negatve) and 0.5(postve) tolerance. The second level decdes y 1 = wth 0.5(negatve) and 0.4(postve) tolerance and y 2 = wth 0.6(negatve) and 0.5(postve) tolerance. The satsfactory soluton (0.6786, , , , , ) can be obtaned by solvng the thrd level auxlary problem usng Lngo. The objectve functon values of the leader and follower are F 1 = and

5 Appl. Math. Inf. Sc. 8, No. 4, (2014) / F 2 = at the satsfactory soluton. The optmal functon values of the leader and follower are all less than those when they make the decson by themselves because the vrtual decson maker who controls the common varable wants to mnmze the sum of the leader and follower s benefts through the common varable. 5 Concluson In ths paper, we propose a b-level lnear programmng problem wth a common varable between the leader and follower. The orgnal b-level problem s transformed nto a tr-level programmng problem by ntroducng a vrtual decson maker, who decdes the common varable to mnmze the sum of the leader and follower s objectve functons. Then, a fuzzy programmng approach s descrbed to solve the transformed problem. The feasblty of the model s llustrated by a numercal example. Acknowledgements The authors would lke to thank the anonymous edtors and revewers for ther nvaluable comments on ths manuscrpt. Ths research s partally funded by the Fundamental Research Funds for the Central Unverstes (Nos. CUG120410, 2011JBM245). References [1] J. F. Bard, Practcal Blevel Optmzaton: Algorthms and Applcatons, Kluwer Academc Publshers, Dordrecht, (1998). [2] B. Colson, P. Marcotte, G. Savard, Blevel programmng: a survey., A Quarterly Journal of Operatons Research, 3, (2005). [3] S. Dempe, Foundatons of Blevel Programmng, Kluwer Academc Publshers, (2002). [4] S. Dempe, Annotated bblography on blevel programmng and mathematcal programs wth equlbrum constrants, Optmzaton, 52, (2003). [5] A. Mgadalas, P. M. Paradalos, P.Värbraud, Multlevel Optmzaton Algorthms and Applcatons, Kluwer Academc Publshers, (1998). [6] Z. Wan, C. Y. Xao, X. J. Wang, et al., Blevel Programmng Model of Optmal Bddng Strateges under the Uncertan Electrcty Markets, Automaton of Electrc Power System, 28, (2004). [7] Je Lu, Chenggen Sh, Guangquan Zhang, On blevel multfollower decson makng: General framework and solutons, Informaton Scences, 176, (2006). [8] C. Sh, J. Lu, G. Zhang, H. Zhou, An extended KuhnCTucker approach for lnear blevel multfollower programmng wth partal shared among followers, IEEE SMC05, The Bg Island, Hawa, USA, submtted for publcaton. [9] Chenggen Sh, Hong Zhou, Je Lu et al, The Kth-best approach for lnear blevel multfollower programmng wth partal shared varables among followers, Appled Mathematcs and Computaton, 188, (2007). [10] R. Jeroslow, The polynomal herarchy and a smple model for compettve analyss. Mathematcal Programmng, 32, (1985). [11] O. Ben-Ayed, C. Blar, Computatonal dffcultes of blevel lnear programmng. Operatons Research, 38, (1990). [12] J. F. Bard, Some propertes of the blevel lnear programmng. Journal of Optmzaton Theory and Applcatons, 68, (1991). [13] L. Vcente, G. Savard, J. Judce, Descent approaches for quadratc blevel programmng. Journal of Optmzaton Theory and Applcatons, 81, (1994). [14] X. Deng, Complexty ssues n blevel lnear programmng. In Mgdalas,A. Pardalos, P. M. and Varbrand,P.(Eds.), Multlevel Optmzaton: Algorthms and Applcatons, Kluwer Academc Publshers, Dordrecht, (1998). [15] H. S. Shh, Y. J. La and E. S. Lee, Fuzzy approach for mult-level programmng problems, Computers & Operatons Research, 23, (1996). [16] H.-S.Shh and E. S.Lee. Compensatory fuzzy multple level decson makng,fuzzy Sets and Systems, 114, (2000). [17] Y. J. La, Herarchcal optmzaton: a satsfactory soluton. Fuzzy Sets and Systems, 77, (1996). [18] E. S. Lee and H.-S.Shh, Fuzzy and mult-level decson makng: an nteractve computaton approach, Sprnger- Verlag London, (2001). [19] E. S. Lee. Fuzzy multple level programmng, Appled Mathematcs and Computaton, 120, (2001). [20] S.Snha, Fuzzy programmng approach to mult-level programmng problems, Fuzzy Sets and Systems, 136, (2003). [21] Z. P. Wan, G. M. Wang, K. L. Hou. An nteractve fuzzy decson makng method for a class of blevel programmng, Book Edtor(s) Ma, J; Yn, YL; Yu, J; et al., Conference 5th Internatonal Conference on Fuzzy Systems and Knowledge Dscovery Locaton Jnan, PEOPLES R CHINA Date, (2008). [22] S. R. Arora, R. Gupta. Interactve fuzzy goal programmng approach for blevel programmng problem, European Journal of Operatonal Research, 194, (2009). [23] S. Dempe. Comment to nteractve fuzzy goal programmng approach for blevel programmng problem by S.R. Arora and R. Gupta, European Journal of Operatonal Research, 212, (2011). [24] A. Snha. Blevel Mult-objectve Optmzaton problem solvng usng progressvely nteractve EMO, Evolutonary Mult-Crteron Optmzaton, (2011). [25] Y. Zheng, Z. P. Wan, G. M. Wang. A fuzzy nteractve method for a class of blevel multobjectve programmng problem, Expert Systems wth Applcatons, 38, (2011). [26] H. Katagr, T. Uno, K. Kato, et al. Random fuzzy blevel lnear programmng through possblty-based value at rsk model, Internatonal Journal of Machne Learnng and Cybernetcs, 1-14 (2012). [27] M. Sakawa, H. Katagr, T. Matsu. Stackelberg solutons for fuzzy random blevel lnear programmng through level

6 1828 L. Zhang: A Fuzzy Algorthm for Solvng a Class of B-Level Lnear... sets and probablty maxmzaton, Operatonal Research, 12, (2012). [28] G. M. Wang, X. J. Wang, Z. P. Wan. A fuzzy nteractve decson makng algorthm for blevel mult-followers programmng wth partal shared varables among followers, Expert Systems wth Applcatons, 36, (2009). [29] G. Zhang, G. Zhang, Y. Gao, J. Lu. A fuzzy blevel model and a PSO-based algorthm for day-ahead electrcty market strategy makng, Knowledge-Based and Intellgent Informaton and Engneerng Systems, (2009). [30] G. Zhang, J. Lu. Fuzzy blevel programmng wth multple objectves and cooperatve multple followers, Journal of Global Optmzaton, 47, (2010). [31] W. F.Balas, M. H.Karwan, Mathematcal methods for multlevel plannng, Reserch Report No. 79-2, (1979). [32] J. F. Bard, Optmalty condton for the b-level programmng problem, WP-104 Unversty of Massachusetts, Bolton, (1983). [33] H. P. Benson, On the structure and propertes of a lnear multlevel programmng problem, Journal of Optmzaton Theory and Applcatons, 60, (1989). [34] E. Lake, Lnear Three-Level Programmng Problem wth The Applcaton to Herarchcal Organzatons, Adds Ababa Unversty, (2007). [35] H. J. Zmmermann, Fuzzy sets and ts applcatons, Kluwer Academc Publshers, Hngham, MA, (1997). [36] Lngo80.hlp, Lngo System Inc., (2003). Lu Zhang s a Ph.D. canddate n Management Scence and Engneerng n the School of Economcs and Management at Chna Unversty of Geoscences (Wuhan). Funded by Chna Scholarshp Councl, she worked n Benedctne Unversty as a vstng scholar from 2012 to Her research nterests nclude Mathematcal Economcs, Organzaton Development, and Agrcultural Economcs and Management.